Black lines on texture
This is most likely caused by your brush being limited to the boundaries of the UV islands. You have 2 ways to fix that :


increase the Bleed value in the Tool Shelf > Options tab
paint on a transparent background and once you're done painting, save the image, import it in the compositor and feed it through an Inpaint node


the latter being the best in complex models and textures but in your case the first is certainly sufficient.

However no matter how simple or complex your model is, you always need margins between your UV islands, even bigger ones if you're using bleed since without a big enough margin it can overlap to another island (which doesn't happen with Inpaint but still a small margin is obligatory).